Advertisement

Circuits, Systems, and Signal Processing

, Volume 35, Issue 3, pp 933–951 | Cite as

Biorthogonal Multiwavelets with Sampling Property and Application in Image Compression

  • Baobin LiEmail author
  • Lizhong Peng
Article
  • 150 Downloads

Abstract

This paper discusses biorthogonal multiwavelets with sampling property. In such systems, vector-valued refinable functions act as the sinc function in the Shannon sampling theorem, and their corresponding matrix-valued masks possess a special structure. In particular, for the multiplicity \(r=2\), a biorthogonal multifilter bank can be reduced to two scalar-valued filters. Moreover, if the vector-valued scaling functions are interpolating, three different concepts: balancing order, approximation order and analysis-ready order, will be shown to be equivalent. Based on this result, we introduce the transferring armlet order for constructing biorthogonal balanced multiwavelets with sampling property. Also, some balanced biorthogonal multiwavelets will be obtained. Finally, application of biorthogonal interpolating multiwavelets in image compression is discussed. Experiments show that for the same length, the biorthogonal multifilter bank is superior to the orthogonal case. Moreover, certain biorthogonal interpolating multiwavelets are also better than the classical Daubechies wavelets.

Keywords

Multiwavelet Biorthogonal multiwavelet  Analysis-ready multiwavelet Transferring armlet order 

Mathematics Subject Classification

42C40a 65T60 15A23 94A08 

Notes

Acknowledgments

The authors thank anonymous reviewers and the editor-in-chief, Prof. M.N.S.Swamy, for their valuable suggestions and comments for improving the presentation of this paper. This work was supported in part by NSFC under Grant No. 11301504 and in part by the President Fund of University of Chinese Academy of Sciences under Grant No. Y25101HY00.

References

  1. 1.
    A. Aldroubi, M. Unser, Families of wavelet transforms in connection with Shannon’s sampling theory and the Gabor transform, in Wavelets: A Tutorial in Theory and Applications, ed. by C.K. Chui (Academic, New York, 1992), pp. 509–528CrossRefGoogle Scholar
  2. 2.
    A. Aldroubi, M. Unser, Sampling procedures in function spaces and asymptotic equivalence with Shannon’s sampling theory. Numer. Funct. Anal. Optim. 15(1–2), 1–21 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    H. Bray, K. McCormick, R.O. Wells, X. Zhou, Wavelet variations on the Shannon sampling theorem. Curr. Mod. Biol. 34(1–3), 249–257 (1995)Google Scholar
  4. 4.
    C.K. Chui, J. Lian, A study of orthonormal multiwavelets. J. Appl. Numer. Math. 20(3), 273–298 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992)CrossRefzbMATHGoogle Scholar
  6. 6.
    J. Geronimo, D. Hardin, P. Massoputs, Fractal functions and wavelet expansions based on several scaling functions. J. Approx. Theory 78(3), 373–401 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    T.N.T. Goodman, C.A. Micchelli, Orthonormal Cardinal Functions, in Wavelets: Theory, Algorithms, and Applications (Academic, San Diego, CA, 1994)Google Scholar
  8. 8.
    Q.T. Jiang, On the design of multifilter banks and orthonormal multiwavelet bases. IEEE Trans. Signal Process. 46(12), 3292–3303 (1998)CrossRefGoogle Scholar
  9. 9.
    Q.T. Jiang, Parametrization of M-channel orthogonal multifilter banks. Adv. Comput. Math. 12(2–3), 189–211 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Q.T. Jiang, Orthogonal and biorthogonal square-root(3)-refinement wavelets for hexagonal data processing. IEEE Trans. Signal Process. 57(11), 4313–14304 (2009)Google Scholar
  11. 11.
    Q.T. Jiang, Biorthogonal wavelets with 4-fold axial symmetry for quadrilateral surface multiresolution processing. Adv. Comput. Math. 34(2), 127–165 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    J. Lebrun, M. vetterli, Balanced multiwavelets, IEEE international conference on acoustics, speech and signal processing, vol. 3 (1997), pp. 2473–2476Google Scholar
  13. 13.
    J. Lebrun, M. Vetterli, High-order balanced multiwavelets: theory, factorization and design. IEEE Trans. Signal Process. 49(9), 1918–1930 (2001)CrossRefMathSciNetGoogle Scholar
  14. 14.
    J.-A. Lian, C.K. Chui, Analysis-ready multiwavelets (armlets) for processing scalar-valued signals. IEEE Signal Process. Lett. 11(2), 205–208 (2004)CrossRefGoogle Scholar
  15. 15.
    B.B. Li, L.Z. Peng, Parametrization for balanced multifilter banks. Int. J. Wavelets Multiresolut. Inf. Process. 6(4), 617–629 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    B.B. Li, L.Z. Peng, Balanced multiwavelets with interpolatory property. IEEE Trans. Image Process. 20(5), 1450–1457 (2011)CrossRefMathSciNetGoogle Scholar
  17. 17.
    B.B. Li, L.Z. Peng, Balanced multifilter banks for multiple description coding. IEEE Trans. Image Process. 20(3), 866–872 (2011)CrossRefMathSciNetGoogle Scholar
  18. 18.
    B.B. Li, L.Z. Peng, Balanced interpolatory multiwavelets with multiplicity \(r\). Int. J. Wavelets Multiresolut. Inf. Process. 10(4), 1250039 (2012)CrossRefMathSciNetGoogle Scholar
  19. 19.
    L. Liu, H. Zhang, Application on Image fusion based on balanced multi-wavelet, 2010 International Symposium on Intelligence Information Processing and Trusted Computing, 512–515 (2010)Google Scholar
  20. 20.
    W. Liu, Z. Ma, X. Tan, Multiple-description video coding based on balanced multiwavelet image transformation. Internet Imaging VI SPIE 5670, 280–291 (2005)CrossRefGoogle Scholar
  21. 21.
    Walid A. Mahmoud, Majed E. Alneby, Wael H. Zayer, 2D-multiwavelet transform 2D-two activation function wavelet network based face recognition. J. Appl. Sci. Res. 6(8), 1019–1028 (2010)Google Scholar
  22. 22.
    M.B. Martin, A.E. Bell, New image compression techniques using multiwavelets and multiwavelet packets. IEEE Trans. Image Process. 10(4), 500–510 (2001)CrossRefzbMATHGoogle Scholar
  23. 23.
    G. Plonka, V. Strela, Construction of multiscaling function’s with approximation and symmetry. SIAM J. Math. Anal. 29(2), 481–510 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    N. Saito, G. Beylkin, Multiresolution representations using the autocorrelation functions of compactly supported wavelets IEEE trans. Signal Process. 41(12), 3584–3590 (1993)zbMATHGoogle Scholar
  25. 25.
    I.W. Selesnick, Interpolating multiwavelet bases and the sampling theorem. IEEE Trans. Signal Process. 47(6), 1615–1621 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    L. Shen, H.H. Tan, J.Y. Tham, Symmetric–antisymmetric orthonormal multiwavelets and related scalar wavelets. Appl. Comput. Harmon. Anal. 8(3), 258–279 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    L. Shen, H.H. Tan, On a family of orthonormal scalar wavelets and related balanced multiwavelets. IEEE Trans. Signal Process. 49(7), 1447–1453 (2001)CrossRefMathSciNetGoogle Scholar
  28. 28.
    G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge, Wellesley, 1996)zbMATHGoogle Scholar
  29. 29.
    V. Strela, P. Heller, G. Strang, P. Topiwala, C. Heil, The application of multiwavelet filter banks to image processing. IEEE Trans. Image Process. 8(4), 548–563 (1999)CrossRefGoogle Scholar
  30. 30.
    P.P. Vaidyanathan, Multirate Systems and Filter Banks, Englewood Cliffs (Prentice Hall, NJ, 1993)Google Scholar
  31. 31.
    Z. Wang, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)CrossRefGoogle Scholar
  32. 32.
    C. Weidmann, J. Lebrun, M. Vetterli, Significance tree image coding using balanced multiwavelets. Proc. ICIP, Chicago, IL, Oct. 1, 97–101 (1998)Google Scholar
  33. 33.
    X.-G. Xia, B.W. Suter, Vector-valued wavelets and vector filter banks. IEEE Trans. Signal Process. 44(3), 508–518 (1996)CrossRefGoogle Scholar
  34. 34.
    X.-G. Xia, Z. Zhang, On sampling theorem, wavelets, and wavelet transforms. IEEE Trans. Signal Process. 41(12), 2535–3524 (1993)Google Scholar
  35. 35.
    J.-K. Zhang, T.N. Davidson, Z.-Q. Luo, K.M. Wong, Design of interpolating biorthogonal multiwavelet systems with compact support. Appl Comput Harmon Anal 11(3), 420–438 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    D.-X. Zhou, Interpolatory orthogonal multiwavelets and refinable functions. IEEE Trans. Signal Process. 50(3), 520–527 (2002)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Computer and ControlUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China

Personalised recommendations